Maximum Power Point Tracking of a DFIG Wind Turbine System 786328456

1. Dissertation
Maximum Power Point Tracking of a
DFIG Wind Turbine System
Graduate School of
Natural Science & Technology
Kanazawa University
Division of
Electrical and Computer Engineering
Student ID No.: 1424042007
Name: Phan Dinh Chung
Chief advisor: Prof. Shigeru YAMAMOTO
Date of Submission: January 6, 2017

3. Abstract
Doubly-fed induction generator (DFIG) has been used popularly in variable speed
wind turbines because the DFIG wind turbine uses a small back-to-back converter
to interface to the connected grid, about 30% comparing to the wind turbine’s ca-
pacity, and provides a control ability as good as a variable speed wind turbine using
a generator with a full converter. The most important purpose of a variable speed
wind turbine or a DFIG wind turbine, in general, is to utilize fully the kinetic en-
ergy of wind for electric generation. To meet this objective, several publications
provided different methods. Generally, the previously proposed schemes can be
listed into two groups including wind speed-based method and wind speed sensor-
less one. With the first group, the wind turbine can give a good performance in
tracking maximum power point but it requires a precise and instantaneous wind
speed measurement; this requirement hardly achieve in practice. With methods in
the second group, an anemometer does not require but the wind turbine using these
methods cannot track maximum power point efficiently under varying wind condi-
tions.
In this dissertation, I proposed two methods and control laws for obtaining max-
imum energy output of a doubly-fed induction generator wind turbine. The first
method aims to improve the conventional MPPT curve method while the second
one is based on an adaptive MPPT method. Both methods do not require any in-
formation of wind data or wind sensor. Comparing to the first scheme, the second
method does not require the precise parameters of the wind turbine. The maximum
power point tracking (MPPT) ability of these proposed methods are theoretically
proven under some certain assumptions. In particular, DFIG state-space models
are derived and control techniques based on the Lyapunov function are adopted to
i

4. derive the control methods corresponding to the proposed maximum power point
tracking schemes. The quality of the proposed methods is verified by the numeri-
cal simulation of a 1.5-MW DFIG wind turbine with the different scenario of wind
velocity. The simulation results show that the wind turbine implemented with the
proposed maximum power point tracking methods and control laws can track the
optimal operation point more properly comparing to the wind turbine using the
conventional MPPT-curve method. The power coefficient of the wind turbine using
the proposed methods can retain its maximum value promptly under a drammactical
change in wind velocity while this cannot achieve in the wind turbine using the con-
ventional MPPT-curve. Furthermore, the energy output of the DFIG wind turbine
using the proposed methods is higher compared to the conventional MPPT-curve
method under the same conditions.
ii

5. 甲（Kou）
様式 ４．
（Form 4）
学 位 論 文 概 要
Dissertation Summary
学位請求論文（Dissertation）
題名（Title） Maximum Power Point Tracking of a DFIG Wind Turbine System
DFIG 風力発電システムの最大電力点追従
専攻（Division）:Electrical and computer engineering
学籍番号（Student ID Number）
：1424042007
氏名（Name）
：Phan Dinh Chung
主任指導教員氏名
（Chief supervisor）
：YAMAMOTO Shigeru
学位論文概要（Dissertation Summary）
Doubly-fed induction generator (DFIG) has been used popularly in variable speed wind turbines because the
DFIG wind turbine uses a small back-to-back converter to interface to the connected grid, about 30% comparing
to the wind turbine’s capacity, and provides a control ability as good as a variable speed wind turbine using a
generator with a full converter. The most important purpose of a variable speed wind turbine or a DFIG wind
turbine, in general, is to utilize fully wind energy for electric generation. To meet this objective, several
publications provided different methods. Generally, the previously proposed schemes can be listed into two
groups such as wind speed-based method and wind speed sensorless one. With the first group, the wind turbine
can give a good performance in tracking maximum power point but it requires a precise and instantaneous wind
speed measurement; this requirement hardly achieve in practice. With methods in the second group, an
anemometer does not require but the wind turbine using these methods cannot track maximum power point
efficiently under varying wind conditions.
In this dissertation, I proposed two methods and control laws for obtaining maximum energy output of
Doubly-fed induction generator wind turbine. The first method aims to improve the conventional MPPT curve
method while the second one is based on an adaptive MPPT method. Both methods do not require any
information of wind data or wind sensor. Comparing to the first scheme, the second method does not require the
precise parameters of the wind turbine. The maximum power point tracking (MPPT) ability of these proposed
methods are theoretically proven under some certain assumptions. In particular, DFIG state-space models are
derived and control techniques based on the Lyapunov function are adopted to derive the control methods
corresponding to the proposed maximum power point tracking schemes. The quality of the proposed methods is
verified by the numerical simulation of a 1.5-MW DFIG wind turbine with the different scenario of wind velocity.
The simulation results show that the wind turbine implemented with the proposed maximum power point tracking
methods and control laws can track the optimal operation point more properly comparing to the wind turbine
using the conventional MPPT-curve method. The power coefficient of the wind turbine using the proposed
methods can retain its maximum value promptly under a drammactical change in wind velocity while this cannot
achieve in the wind turbine using the conventional MPPT-curve. Furthermore, the energy output of the DFIG
wind turbine using the proposed methods is higher compared to the conventional MPPT-curve method under the
same conditions.

6. Contents
Abstract i
Contents iv
List of Figures v
List of Tables vii
Acknowledgement ix
1 Introduction 1
1.1 Outline of The Dissertation . . . . . . . . . . . . . . . . . . . . . . 4
2 DFIG-Wind Turbine 5
2.1 Wind turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 DFIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Controller Design and Maximum Power Strategy 15
3.1 Maximum power point tracking . . . . . . . . . . . . . . . . . . . 15
3.2 Design RSC controller for improved MPPT scheme . . . . . . . . . 19
3.2.1 RSC Controller for power adjustment . . . . . . . . . . . . 19
3.2.2 Improved MPPT scheme . . . . . . . . . . . . . . . . . . . 20
3.3 Design RSC controller for adaptive MPPT scheme . . . . . . . . . 21
3.3.1 RSC controller for rotor speed adjustment . . . . . . . . . . 21
3.3.2 Adaptive MPPT scheme . . . . . . . . . . . . . . . . . . . 22
3.4 Comparison of two proposed MPPT schemes . . . . . . . . . . . . 24
iii

7. 4 Simulation and Discussions 27
4.1 Parameters for improved MPPT method . . . . . . . . . . . . . . . 30
4.2 Parameters for adaptive MPPT method . . . . . . . . . . . . . . . . 30
4.3 Simulation results and disscusion . . . . . . . . . . . . . . . . . . . 31
5 Conclusion 39
A DFIG Wind Turbine 41
A.1 Proof of Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
B Controller Design and Maximum Power Strategy 43
B.1 Proof of Lemma 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 43
B.2 Proof of Lemma 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
B.3 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 44
B.4 Proof of Lemma 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
B.5 Matrix inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
B.6 Proof of Lemma 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
B.7 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Publications 55
Bibliography 56
iv

8. List of Figures
Fig. 1.1 Variable speed wind turbine based on: (a) full power converter
and (b) partial power converter. . . . . . . . . . . . . . . . . . . . . 2
Fig. 2.1 Overall system of the doubly-fed induction generator (DFIG)
wind turbine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Fig. 2.2 Characteristic of wind turbine: (a) Cp versus λ and (b) Pm ver-
sus ωr at different wind speeds as β = 0. . . . . . . . . . . . . . . . 9
Fig. 3.1 Wind turbine characteristic of (3.4) for β = 0: (a) Cp(λ), (b)
Pm(λ, Vw) and Pmppt(ωr), and (c) contour of wind turbine. . . . . . . 17
Fig. 3.2 Control diagram using the improved MPPT method. . . . . . . 21
Fig. 3.3 Control diagram using the adaptive MPPT method. . . . . . . . 24
Fig. 4.1 ζp(ωr, Vw) (a),
δ3
δ2
(b), and ζ(ωr, Vw). . . . . . . . . . . . . . . . 29
Fig. 4.2 Wind speed profile: (a) wind speed and (b) wind acceleration. . 32
Fig. 4.3 Simulation results: (a) ωr(t)−ωropt(Vw(t)), (b) power coefficient
Cp(λ(t)), (c) Pmax(t) − Pm(t), and (d) electrical energy output. . . . . 35
Fig. 4.4 Simulation results: (a) ratio k̂opt/kopt and (b) ωropt(t) − ω̂ropt(t),
(c) irdref(t) − ird(t), (d) xr(t) − xPQ(t). . . . . . . . . . . . . . . . . . 36
Fig. 4.5 Simulation results: (a) wind speed, (b) power coefficient, (c)error
Pmax(t) − Pm(t), and (d) energy output. . . . . . . . . . . . . . . . . 37
Fig. 4.6 Simulation results: (a) wind speed, (b) power coefficient, (c)
error Pmax(t) − Pm(t), and (d) energy output. . . . . . . . . . . . . . 38
v

9. vi

10. List of Tables
Table 3.1 Comparision of two MPPT methods . . . . . . . . . . . . . . 25
Table 4.1 Parameters in simulations (DFIG[25] and wind turbine) . . . 28
vii

11. viii

12. Acknowledgement
Firstly, I would like to express my thanks to Prof. Shigeru YAMAMOTO who
has opened a chance for me to pursue this doctoral course and helped me to effec-
tuate this research. I got a variety of new knowledge and experience from Prof and
his lab. I also want to thank Prof. Osamu KANEKO who supported me many things
in research.
Secondly, I want to say thank you for Vietnam International Education Develop-
ment - Ministry of Education and Training who support financially for me to pursue
this research. I also want to say thank you to Dr. Tran Vinh Tinh and Associate Prof.
Dinh Thanh Viet, leaders of Electrical engineering Department-Danang University
of Science and Technology, who gave me time for this PhD course.
Next, I would like to say thank you to all members of my family, parents and
siblings, who always encourage me to study. In particular, thanks for the help of
my brothers and sisters in taking care my parents, I can be assured to concentrate
on studying.
Last but not least, I want to thank you for sharing knowledge from all members
of MoCCoS laboratory, Mr. Mohd Syakirin, Mrs. Dessy Novita, and Mr.Herlambang
Saputra. I want to thank Mr. Kyohei Asai, my tutor, who help me to get on well in
daily life in Kanazawa. By the way, I also thank all friends who shared everything
in daily life.
Phan Dinh Chung
Kanazawa, January 2017
ix

13. x

14. Chapter 1
Introduction
The fossil-fuel exhaustion and environment pollution concerns have urged researchers
to utilize renewable energy resources such as the wind, solar, and wave energy for
generating electricity. Until now, the use of wind energy for electric generation has
been developing in many countries; many large-scale wind farms, both offshore and
onshore, have been built and exploited. According to Global Wind Energy Coun-
cil (GWEC) [1], over 80 countries in the world have been utilizing wind energy
for electricity generation with installed wind capacity in total up to 433GW at the
end of 2015 and 14 of these countries, installed capacity in total is over 5000MW.
Referring to the prediction of GWEC, the total wind capacity data will increase to
792GW at the end 2020. We always expect that electric energy withdrawing from a
wind turbine system should be as high as possible. In other words, we should utilize
fully wind energy for electric generation from a wind turbine.
To optimally utilize wind energy, the energy conversion efficiency of wind tur-
bines must reach the utmost limit. Therefore, maximum power point tracking
(MPPT) is an essential target in wind turbine control. To track the maximum power
point, the rotor speed of the wind turbine/generator should be adjustable. Hence,
the concept of a variable-speed wind turbine (VSWT) was proposed. According
to [2, 3, 4, 5], when the generator in a VSWT operates at variable speeds, its out-
put is often synchronized with the grid via a converter system. Depending on the
type of generator used in the VSWT, the converter’s size will vary as shown in
Fig.1.1. With VSWT based on synchronous generator (SG), permanent magnetic
1

15. synchronous generator (PMSG), or squirrel-cage induction generator (SCIG), the
generator is interfaced to grid through a converter as Fig.1.1a; during operation,
all active power generated by the generator is transfered to the connected grid and
hence, a full converter whose capacity should not be smaller than the generator-
wind turbine’s capacity is used [3, 4, 5]. However, for VSWT that use a doubly fed
induction generator (DFIG) in Fig.1.1b, a partial converter is required on the rotor
side [2] because the power generated/absorbed on the rotor side is around 30% of
the DFIG-wind turbine capacity. In other words, compared to a full converter-based
VSWT, the use of a DFIG wind turbine is more economical; in fact, DFIG wind tur-
bines are more frequently used in large wind farms. Therefore, control for a MPPT
target in DFIG-based wind turbines has become an interesting topic.
(a)
(b)
Fig. 1.1: Variable speed wind turbine based on: (a) full power converter and (b)
partial power converter.
To track the maximum power point during operation, a wind turbine must be
generally equipped with a good controller integrated with a comprehensive MPPT
algorithm. Many MPPT methods have been proposed [5, 6, 7, 8, 9, 10, 11, 12, 13].
Original methods are based on the characteristic curve. They use the curve of the
optimal power versus wind speed, for example, or the optimal tip-speed ratio of a
2

16. wind turbine and wind data to determine the reference signal for the controller [8, 6].
These methods are called wind-data-based methods. Generally, with wind-data-
based methods, the MPPT ability of a wind turbine is appreciably high if accurate
wind data is available. However, because of the rapid natural fluctuation of wind,
wind speed measurement is hardly reliable [14]. To overcome this drawback, other
methods such as the MPPT-curve method [11, 12, 10, 13] and perturbation and
observation (P&O) method [7] were suggested. They operate basically on the output
of the generator; hence, they are called wind speed-sensorless methods. Compared
to the wind-data-based methods, the wind speed-sensorless methods cannot track
the optimum point as efficiently as [15]. However, this method is often implemented
in wind turbines because there is no requirement for an anemometer. The P&O
method is originally applied for extremum seeking in small inertia systems such
as photovoltaic power systems or small-size PMSG wind turbines with a DC/DC
converter [5, 7]. Unlike the P&O method, the MPPT-curve method, which indexes
the current power output (or rotor speed) as well as the wind turbine’s MPPT curve
to determine the reference rotor speed (or power output) [11, 12, 13], can apply to
both large- and-small scale wind turbines; it is more efficient and does not require
any perturbation signal [8]. However, for the high inertia of a generator wind turbine
system, a wind turbine using the MPPT-curve method cannot track the maximum
point as rapidly as a wind turbine using the wind-data-based method [15].
In terms of designing the controller for a wind turbine, traditional proportional-
integral (PI) control is used for many purposes, including rotor-speed, current, and
power control [11, 12]. A drawback of PI control is that stability is not theoretically
guaranteed [16, 17]. Thus, sliding-mode control has been recently developed [18,
19, 20, 21, 22]. In fact, sliding-mode control has been applied to the rotor speed
[20, 21, 22]. However, wind speed measurement is prerequisite for sliding mode
control.
This research suggests two new schemes to maximize the energy output of a
DFIG wind turbine without any information about the wind data or an available
anemometer. These proposed schemes are based on the improvement of the wind
turbine’s MPPT curve and the adaptation of MPPT curve; their names are improved
3

17. MPPT-curve method and adaptive MPPT method. Certainly, the improved MPPT-
curve method is completely independent to the adaptive MPPT method. In addition
to the proposed MPPT methods, in this research, two new controllers based on Lya-
punov control theory will be designed for power and rotor speed adjustment pur-
poses, corresponding to the improved and adaptive MPPT method. The efficiency
of the proposed schemes will be verified, analyzed, and compared with the con-
ventional MPPT curve method with PI controllers by the simulation of a 1.5-MW
DFIG wind turbine in a MATLAB/Simulink environment.
1.1 Outline of The Dissertation
This dissertation is organized as follows.
In Chapter 2: DFIG-wind turbine
In Chapter 3: Controller design and proposed MPPT schemes
In Chapter 4: Simulation results and discussion
In Chapter 5: Conclusion
Appendix A
Appendix B
4

18. Chapter 2
DFIG-Wind Turbine
As we mentioned in Chapter 1, DFIG-wind turbines are produced popularly in mid-
dle and large capacity scale, from several hundred kilowatts to several megawatts.
A DFIG wind turbine consists of two main components including a wind turbine
and doubly fed induction generator (DFIG) [23]. The wind turbine aims to absorb
kinetic energy from wind and produces mechanical torque to rotate its shaft. In
normal operation, the wind turbine’s velocity range is several rpm [24] while the
DFIG’s rotational speed varies from several hundred to thousand rpm depending on
the number of pole pairs. Hence, the wind turbine is linked to the DFIG through a
shaft system including a turbine shaft, generator shaft, and gearbox (GB), as shown
in Fig.2.1. Through this shaft system, the mechanical energy on the turbine shaft is
transferred to the generator in order to convert to electrical energy. In this research,
this shaft system is a union mass. It means there is not energy losses on the shaft
system. Generally, the dynamic equation for a generator-wind turbine system [18]
is used to described
J
d
dt
ωr(t) = Tm(t) − Te(t), (2.1)
where, J is the inertia of the generator-wind turbine system; ωr is the rotor speed
of the wind turbine; Tm and Te respectively stand for the mechanical torque of the
wind turbine and the electrical torque of the generator referring to the turbine speed.
Moreover, to use the mechanical and electrical power Pm and Pe, respectively,
5

19. we can rewrite (2.1) as
Jωr(t)
d
dt
ωr(t) = Pm(t) − Pe(t). (2.2)
Fig. 2.1: Overall system of the doubly-fed induction generator (DFIG) wind turbine.
The purpose of the DFIG is to convert the mechanical energy on its shaft to
electric energy on both stator and rotor winding. Generally, DFIG is an induction
generator like a SCIG but its rotor winding is not shorted-circuit and is connected
to an external grid. During operation, the DFIG must receive reactive power from
external grid through the stator or rotor winding to produce flux in the DFIG; the
electric frequency on the stator winding depends on the frequency of the external
grid while the electric frequency on the rotor side depends on the generator’s rota-
tional speed. Hence, when the DFIG-wind turbine operates in variable speed, the
frequency on the rotor side varies among wind speed; to interface to a connected
grid, the stator side of the DFIG is often connected directly to the grid whereas
the rotor side is connected through a back-to-back converter. The main objective
of the back-to-back converter is to synchronize between the rotor winding and the
6

20. connected grid. General speaking, the back-to-back converter includes a rotor-side
converter (RSC), grid-side converter (GSC), and DC link. In the DFIG-wind tur-
bine, since the rotor side can supply/absorb active power to/from the connected grid,
the RSC (GSC) can work as either a rectifier (inverter) or an inverter (rectifier). This
is reason why a bidirectional or back-to-back converter using IGBT-valves must be
installed on the rotor side. Furthermore, the slip speed of the DFIG is only from
-30% to 30% so the maximum power flowing through the back-to-back converter is
around 30% of DFIG capacity. Therefore, the converter capacity is around 30% of
DFIG capacity and this is a good point of DFIG-wind turbine.
For the DFIG wind turbine, thanks to the back-to-back converter, the DFIG-
wind turbine gives a qualified control ability. Through the RSC, we can adjust the
turbine speed to achieve a desired power and adjust both reactive power quantity
and direction on the stator side. The GSC is often controlled to remain a constant
DC voltage on the DC link and to support reactive power to the connected grid. To
be more convenient in designing controllers, a dq frame is often used and all three
phase signals such as current and voltage are converted from an abc frame to the dq
frame. Hence, the output of the controllers must be converted from the dq frame
to the abc frame, in Fig.2.1, and then pulse-wide-modulations (PWMs) generate
pulses to switch on/off IGBT-valves in RSC and GSC.
In this research, we only focus on designing the RSC controller and propos-
ing MPPT method. Therefore, in this chaper, we only describe the mathematical
modeling of the wind turbine and the DFIG.
2.1 Wind turbine
As we know, the wind turbine is to absorb kinetic from wind through its blade
system and convert it to mechanical energy on its shaft. When the turbine rotates at
ωr and wind speed is Vw, the tip speed ratio is defined by
λ(ωr, Vw) ,
Rωr
Vw
, (2.3)
7

21. where R is the length of its blade. Mechanical power on its shaft Pm is written as
Pm(λ, Vw) ,
1
2
ρπR2
Cp(λ, β)V3
w, (2.4)
where ρ, and Cp(λ, β) are the air density, and power coefficient, respectively. The
power coefficient Cp represents the energy conversion efficiency of the wind turbine
and generally depends on both the tip speed ratio λ and the pitch angle β. Normally,
the Cp(λ, β) depends on the manufacture of the wind turbine and it has a maximum
point respecting to λ at a constant β.
By using (2.3), (2.4) becomes
Pm(ωr, Vw) =
1
2
ρπR5 Cp(λ, β)
λ3
ω3
r . (2.5)
For example, a 1.5MW wind turbine with
Cp(λ, β) = (
79.254
λi
− 1.5β − 5.315)e−21/λi
+ 0.009λ, (2.6)
1
λi
=
1
λ + 0.08β
+
0.035
β3 + 1
, (2.7)
has Cp(λ, β) and Pm at different wind speeds as β = 0 as shown in Fig.2.2. From
Fig.2.2a, we can see that at a constant β, Cp(λ) has an unique maximum point
respecting to λ. At a constant λ, when β increases the Cp(β) decreases. Fig.2.2b
shows the mechanical power Pm versus ωr as β = 0. Obviously, Fig.2.2b can be
divided into three regions:
• Parking region
When the wind speed is below cut-in speed Vw−cut−in, the wind turbine is
locked and the wind turbine system does not operate.
When Vw = Vw−cut−in, the wind turbine begins the star-up process. Since the
kinetic energy from the wind is insignificant, the wind turbine is controlled at
constant speed until the wind speed is equal to a minimum speed Vw min. This
period, Vw−cut−in ≤ Vw ≤ Vwmin, is considered as the start-up interval of the
wind turbine.
• Optimal power control region
8

22. (a)
(b)
Fig. 2.2: Characteristic of wind turbine: (a) Cp versus λ and (b) Pm versus ωr at
different wind speeds as β = 0.
When the wind speed is between Vw min and the rated speed Vwrated, Vw min ≤
Vw ≤ Vwrated, the Pm (2.4) always fails to be over its rated value, even though
Cp is maximum value. Therefore, to utilize the wind energy optimally, the
power coefficient of the wind turbine Cp should be maximum; it means the
Pm is maximum. Hence, when the wind turbine operates in the region of
Vw min ≤ Vw ≤ Vwrated, the wind turbine is controlled to extract optimal power
and this region is called optimal power control region.
From Fig. 2.2a, to get maximum Cp, the pitch angle β should be keep at
zero and the wind turbine speed should be adjusted so that λ approaches λopt.
This makes Cp(λ, 0) to reache maximum value and then the Pm approaches its
maximum value.
9

23. • Pitch control region
When the wind speed is over the rated speed Vwrated = 12m/s, the Pm is over
the rated value and this is very dangerous to all equipments in the wind turbine
system. In this case, the wind turbine must be controlled to reduce Cp(λ, β)
and as a result, the Pm can be limited at the rated value. From Fig. 2.2a,
to decrease Cp(λ, β), we must increase pitch angle β. Hence, when the wind
turbine operates in the region of Vw > Vwrated, the pitch system has to be
controlled and this region is called pitch contrl region.
This research, we propose two independent methods such that the wind turbine
extracts maximum mechanical power as Vw min ≤ Vw ≤ Vwrated. Hence, throughout
this paper, we fix β as a constant and we simply denote it as Cp(λ).
From (2.3), we can regard Pm as
Pm(ωr, Vw) =
1
2
ρπR2
Cp(λ(ωr, Vw))V3
w. (2.8)
2.2 DFIG
In the dq frame, the DFIG can be described as [25, 26]









vs(t) = Rsis(t) + Ls
d
dt
is(t) + Lm
d
dt
ir(t) + ωsΘ(Lsis(t) + Lmir(t))
vr(t) = Rrir(t) + Lr
d
dt
ir(t) + Lm
d
dt
is(t) + ωss(t)Θ(Lmis(t) + Lrir(t))
(2.9)
where vs =
vsd vsq
is the stator-side voltage, vr =
vrd vrq
is the rotor-side
voltage, is =
isd isq
is the stator-side current, ir =
ird irq
is the rotor-side
current and Θ =










0 −1
1 0










. ω, R, L and s represent rotational speed, resistance,
inductance and rotor slip, respectively; subscripts r, s and m stand for rotor-side,
stator-side and magnetization. Note that ωs is normally constant.
The rotor slip of the DFIG is defined by:
s(t) , 1 −
ωr(t)
ωs
. (2.10)
10

24. Assumption 1. The stator flux is constant, and the d-axis of the dq-frame is oriented
with the stator flux vector. Hence,
Ψs(t) =










Ψsd(t)
Ψsq(t)










≡










Ψsd
0










= Lsis(t) + Lmir(t). (2.11)
Then,
Ls
d
dt
is(t) + Lm
d
dt
ir(t) = 0. (2.12)
Moreover, the resistance of the stator winding can be ignored, i.e., Rs = 0.
Lemma 1. Under Assumption 1, in a DFIG (2.9), the stator-side voltage becomes
constant as
vs(t) =
0 Vs
, (2.13)
where Vs is the magnitude of the stator voltage kvs(t)k. Moreover, the rotor-side
current ir and voltage vr satisfy
d
dt
ir(t) = Ai(t)ir(t) + σ−1
vr(t) + di(t), (2.14)
where
σ , Lr −
L2
m
Ls
, (2.15)
Ai(t) ,










−σ−1
Rr ωss(t)
−ωss(t) −σ−1
Rr










, (2.16)
di(t) , −
Lm
Lsσ
s(t)










0
Vs










. (2.17)
Proof. By substituting (2.12) and Rs = 0 into (2.9), we have (2.13) as
vs(t) =










0
ωsΨsd










=










0
Vs










, (2.18)
because Vs = kvs(t)k = |ωsΨsd|. From (2.11) and (2.18), we have
Lsis(t) + Lmir(t) =
1
ωs










Vs
0










. (2.19)
11

25. Hence,
is(t) = −
Lm
Ls
ir(t) +
1
Lsωs










Vs
0










, (2.20)
d
dt
is(t) = −
Lm
Ls
d
dt
ir(t). (2.21)
Substituting (2.20) and (2.21) into the second equation of (2.9), we have
vr(t) = Rrir(t) + Lr −
L2
m
Ls
!
d
dt
ir(t) + ωss(t)Θ
Lm
Lsωs










Vs
0










+ ωss(t)Θ Lr −
L2
m
Ls
!
ir(t). (2.22)
From (2.10) and (2.22), we obtain (2.14).
The power in the stator side are given as [26]
xpq(t) ,










Qs(t)
Ps(t)










=










−isq(t) isd(t)
isd(t) isq(t)




















vsd(t)
vsq(t)










. (2.23)
By substituting (2.18) and (2.19) into (2.23), we have
xpq(t) = Vsis(t) = −Ṽsir(t) +
1
Lsωs










V2
s
0










, (2.24)
where Ṽs =
Lm
Ls
Vs. When the power loss in the DFIG can be neglected, the power
output of the generator Pe is described by
Pe(t) = Ps(t) + Pr(t) = (1 − s(t)) Ps(t), (2.25)
where Pr is the rotor-side active power. Hence,
Te(t) =
Pe(t)
ωr(t)
=
pnNPs(t)
ωs
= −
pnNLm
Lsωs
Vsirq(t), (2.26)
where, pn and N are the number of pole pairs and gearbox ratio, respectively.
Lemma 2. A state-space representation of the active power Ps and the reactive
power Qs on the stator side is given by
d
dt
xpq(t) = Ai(t)xpq(t) −
Ṽs
σ
vr(t) + cpq(t), (2.27)
12

26. where
cpq(t) = −
Ai(t)
Lsωs










V2
s
0










− Ṽsdi(t) =
V2
s
σLs











Rr
ωs
Lr s(t)











.
In addition, under Assumption 1, a state-space representation of the DFIG from
(2.9) is described by
d
dt
xPQ(t) = APQ(t)xPQ(t) + BPQ(t)vr(t) + dPQ(t), (2.28)
where
xPQ(t) =










Qs(t)
Pe(t)










, (2.29)
APQ(t) = C(t)−1
Ai(t)C(t) − Ċ(t)
=



















−1 −
Rr
σ
ω2
s
ωr(t)
s(t)
−ωr(t)s(t)
d
dt
ωr(t) − ωs
Rr
σ
ωr(t)



















, (2.30)
BPQ(t) = −
Ṽs
σ
C−1
(t), C(t) =











1 0
0
ωs
ωr(t)











, (2.31)
dPQ(t) = C(t)−1
cpq(t) =
V2
s
σLsωs










Rr
Lrωr(t)s(t)










. (2.32)
Proof. See A.1 for the proof of Lemma 2.
13

27. 14

28. Chapter 3
Controller Design and Maximum
Power Strategy
The main objective of this chapter proposes two new schemes for tracking maxi-
mum power point, including improved MPPT scheme and adaptive MPPT scheme,
when the wind turbine operates in the optimal power control region. The improved
MPPT scheme is independent to the adaptive MPPT scheme. In addition to these
schemes, we design two RSC controllers corresponding to these schemes. These
RSC controllers are independent together. For the improved MPPT scheme, we de-
sign the RSC controller for the power adjustment. For the adaptive MPPT scheme,
the RSC controller is designed to adjust the rotor speed and current. Hereafter,
following assumptions are used.
Assumption 2. We can measure ir, is, ig, vs, and ωr. In addition, we know parame-
ters Rr, Ls, Lr and Lm and manipulate Qs, Pe.
Assumption 3. The dq/abc transformation block, the PWM and the IGBT-valves
in RSC in Fig.2.1 operate properly.
3.1 Maximum power point tracking
The optimal power control region D of the wind turbine is defined by
D , {(ωr, Vw) | ωrmin ≤ ωr ≤ ωrrated, Vwmin ≤ Vw ≤ Vwrated, and Cp(λ) 0}, (3.1)
15

29. where, ωrmin and ωrrated are the minimum and rated rotor speed; Vwmin and Vwrated are
the minimum and rated wind speed. In this region, the tip-speed ratio is bounded as
λmin ,
Rωrmin
Vwrated
≤ λ(t) ≤ λmax , max{λ | Cp(λ) 0}.
This research aims to suggest MPPT schemes and controllers such that the wind
turbine can work in the optimal power control region D of the MPPT curve.
MPPT-curve method
We consider a maximization of the mechanical power to change ωr. Evidently, it is
equivalent to a maximization of Cp(λ(ωr, Vw)). That is,
Cpmax , Cp(λopt), (3.2)
λopt , arg max
λ
Cp(λ). (3.3)
In this research, we use
Cp(λ) =
165.2842λ−1
− 16.8693
e−21λ−1
+ 0.009λ (3.4)
for the pitch angle β = 0. It has an unique maximum point of Cpmax = 0.4 at
λopt = 6.7562, as shown in Fig.3.1a.
The optimal rotor speed
ωopt(Vw) ,
λoptVw
R
, (3.5)
achieves the maximal mechanical power
max
ωr
Pm(ωr, Vw) =
1
2
ρπR2
CpmaxV3
w = koptω3
opt(Vw), (3.6)
kopt ,
1
2
ρπR5 Cpmax
λ3
opt
. (3.7)
To maximize the mechanical power, if we have a wind speed Vw, we simply control
to make ωr(t) track the ωopt(Vw(t)) given in (3.5). However, since it is difficult to
obtain precise values of Vw, we generally control ωr or Pe to make the mechanical
power Pm(ωr, Vw) track
Pmppt(ωr) = koptω3
r , (3.8)
16

30. instead of (3.6). Pmppt(ωr) is a locus of the peak of Pm(ωr, Vm) as Vm changes in the
optimal power control region D Fig.3.1b. This is called the MPPT-curve method or
MPPT scheme.
(a)
(b)
(c)
Fig. 3.1: Wind turbine characteristic of (3.4) for β = 0: (a) Cp(λ), (b) Pm(λ, Vw) and
Pmppt(ωr), and (c) contour of wind turbine.
17

31. Note that the optimal control region D is divided into three parts as Fig. 3.1c
Dlr ,
n
(ωr, Vw) ∈ D | Rωr/Vw λopt
o
(3.9)
Dopt ,
n
(ωr, Vw) ∈ D | Rωr/Vw = λopt
o
(3.10)
Dul ,
n
(ωr, Vw) ∈ D | Rωr/Vw λopt
o
. (3.11)
Moreover, from Fig. 3.1a and Fig. 3.1c, we have
∂
∂λ
Cp(λ)





















0 (ωr, Vw) ∈ Dlr
= 0 (ωr, Vw) ∈ Dopt
0 (ωr, Vw) ∈ Dul.
(3.12)
In the analysis of the proposed scheme, we use
ζ(ωr, Vw) , −
Pm(ωr, Vw) − Pmppt(ωr)
ωr(ωr − ωropt(Vw))
. (3.13)
By using L’Hôpital’s rule, we ensure that
lim
ωr→ωropt(Vw)
ζ(ωr, Vw) = − lim
ωr→ωropt(Vw)
d
dωr
Pm(ωr, Vw) −
d
dωr
Pmppt(ωr)
d
dωr
(ωr(ωr − ωropt(Vw)))
= − lim
ωr→ωropt(Vw)
R
Vw
∂
∂λ
Pm(λ, Vw) − 3koptω2
r
2ωr − ωropt(Vw)
=
3koptωropt(Vw)2
ωropt(Vw)
= 3koptωropt(Vw). (3.14)
By regarding (2.8) to use (2.3) as
Pm(ωr, λ) =
1
2
ρπR5 Cp(λ)
λ3
ω3
r (3.15)
and from Pmppt(ωr) = koptω3
r and the definition of kopt in (3.7), we have
Pm(ωr, Vw) − Pmppt(ωr) =
1
2
ρπR5
ω3
r







Cp(λ)
λ3
−
Cpmax
λ3
opt






 . (3.16)
Hence,
ζ(ωr, Vw) =
ρπR5
ω2
r
2(ωr − ωropt(Vw))







Cpmax
λ3
opt
−
Cp(λ)
λ3






 (3.17)
=
ρπR6
ω2
r
2Vwλ3(λ − λopt)







Cpmax
λ3
opt
λ3
− Cp(λ)






 . (3.18)
18

32. Lemma 3. For a point (ωr, Vw) ∈ Dopt ∪ Dlr, ζ(ωr, Vw) 0. Moreover, for a point
(ωr, Vw) ∈ Dul, if
Cpmax
λ3
opt
−
Cp(λ)
λ3
0, (3.19)
then
ζ(ωr, Vw) 0. (3.20)
Proof. See B.1 for the proof of Lemma 3.
Remark 1. From (3.8) and (3.13), we can write
Pm(t) − koptω3
r (t) = −ζp(Vw, ωr)
λ(t) − λopt
, (3.21)
where
ζp(Vw, ωr) = ζ(Vw, ωr)
Vw(t)ωr(t)
R
, (3.22)
and ζp(Vw, ωr) is also positive, continuous and bounded in the region D.
Assumption 4. When the wind turbine operates in the region D,

37. d
dt
ωropt(Vw(t))

42. =
λopt
R
d
dt
Vw(t) ≤
λopt
R

47. d
dt
Vw(t)

52. , γ. (3.23)
3.2 Design RSC controller for improved MPPT scheme
This section, we will design a control law for the RSC to adjust the power output
of the DFIG and an improved MPPT method which develop from the MPPT-curve
method.
3.2.1 RSC Controller for power adjustment
The objective of RSC is to maintain, at the desired references, the reactive power in
the stator side Qs and the total active power output Pe of the DFIG,
x
r =
Qsref Peref
. (3.24)
From (2.28), to adjust Qs and Pe, vrd and vrq must be regulated, respectively. To
perform this task, previous studies employed PI controllers [12, 27]. Hereafter, a
new control law is proposed.
19

53. Lemma 4. Under Assumptions 1 and 2, when we can measure d
dt
ωr(t) for any
desired reference xr from (3.24), if we use any positive definite matrix P,
vr(t) = −BPQ(t)−1
APQ(t)xPQ(t) + P(xr(t) − xPQ(t)) −
d
dt
xr(t) + dPQ(t)
!
(3.25)
for the DFIG (2.28), then it is ensured that
lim
t→∞
(xr(t) − xPQ(t)) = 0. (3.26)
Proof. See B.2 for the froof of Lemma 4.
From Lemma 4, if the rotor voltage vr is designed as (3.25), the power output
of DFIG will converge to its reference value given by (3.24).
3.2.2 Improved MPPT scheme
The main objective of this subsection is to propose a new MPPT scheme that im-
proves the conventional MPPT-curve method [12] so that Pm approaches the neigh-
bor of Pmax as ωrmin ≤ ωr(t) ωrrated. From (3.5) and (3.6), Pm(t) only reaches
the neighbor of Pmax as ωr approaches the neighbor of ωropt or λ approaches the
neighbor of λopt. Therefore, in this subsection, a new strategy is proposed, such that
λ approaches the neighbor of λopt.
Theorem 1. Under Assumption 4, suppose that we use a positive constant α J,
kopt in (3.7), and Peref in (3.24) for the RSC control (3.25) as
Peref(t) = koptω3
r (t) − αωr(t)
d
dt
ωr(t), (3.27)
if there exists a positive constant χ, such that
P̃ := 2P −












0 0
0 χ−1 λ(t)
(J − α)ω2
r (t)












0 (3.28)
for the definite matrix P 0 in (3.25) and all t, then there exists a time t0 0, such
that

56. λ(t) − λopt

59. ≤ 2(J − α)γ
R
λopt
max
ω2
r (t)
(2ζp(t) − χ)Vw(t)
, (3.29)
for all t ≥ t0.
20

60. Fig. 3.2: Control diagram using the improved MPPT method.
Proof. See B.3 for the proof of Theorem 1.
From Theorem 1, if the reference power Peref in (3.24) is calculated as (3.27),
the tip-speed ratio λ of the wind turbine will converge to the neighbor of λopt and,
hence, Pm will approach its maximum value. Hence, we have control diagram as
Fig. 3.2.
Remark 2. By multiplying two side of (3.29) by
Vw
R
, we have

63. ωr(t) − ωropt(t)

66. =

69. λ(t) − λopt

72. Vw
R
≤ 2(J − α)γ
Vwrated
λopt
max
ω2
r (t)
(2ζp(t) − χ)Vw(t)
.
(3.30)
3.3 Design RSC controller for adaptive MPPT scheme
In this section, I will propose a new control law for the RSC to adjust the rotor speed
and a new MPPT method which is named adaptive MPPT scheme. Noted that the
RSC controller and adaptive MPPT scheme in this section are independent to the
RSC controller and improved MPPT scheme which were designed in Section 3.2.
3.3.1 RSC controller for rotor speed adjustment
The objective of RSC is to maintain the ird of the DFIG and the rotor speed ωr of
the wind turbine at the desired references. From (2.1), (2.14), and (2.26), to control
ird and ωr, we can adjust ird and irq by vrd and vrq, respectively. To achieve this task,
in previous research, traditional PI control was used [12, 28, 29, 30, 31]. In this
research, a new law for rotor speed control is proposed.
21

73. Lemma 5. Under Assumptions 2 and 3, for any reference irdref and ωrref, if vr of the
DFIG (2.9) is designed as
vr(t) = σ(−Ai(t)ir(t) − di(t) +
d
dt
irref(t) + K (irref(t) − ir(t))), (3.31)
where, for kd 0,
irref(t) ,










irdref(t)
irqref(t)










=










irdref(t)
irq(t) + kd
d
dt
(ωrref(t) − ωr(t)) + kp(ωrref(t) − ωr(t))










, (3.32)
and if the feedback gain K and kp satisfy
Q̃ ,


















2kp
0 −1










0
−1










K
+ K


















0, (3.33)
then
lim
t→∞
(irref(t) − ir(t)) = 0, and lim
t→∞
(ωrref(t) − ωr(t)) = 0. (3.34)
Proof. See B.4 for the proof of Lemma 5.
Hence, from Lemma 5 and Assumption 3, if the rotor-side voltage of the DFIG
is adjusted to satisfy (3.31), ir(t) and ωr(t) will converge to the desired values irref(t)
and ωrref(t), respectively.
3.3.2 Adaptive MPPT scheme
In this subsection, we propose a new MPPT scheme using no real-time informa-
tion about Vw(t). The scheme aims to reduce |ωropt(Vw(t)) − ωr(t)| to achieve the
maximum P(ωr, Vw).
Assumption 5. The precise value of kopt for the MPPT curve is not available. In-
stead, we can use the estimate k0
opt with
k0
opt = (1 + δ)kopt, |δ| ≤ δmax. (3.35)
22

74. The proposed MPPT scheme is given as the reference ωrref in (3.32) for the RSC
control (3.31) as
ωrref(t) ,






P̂mppt(t)
k̂opt(t)






1/3
, (3.36)
P̂mppt(t) = ωr(t) k1
d
dt
ωr(t) − k2
ωr(t) − ω̂ropt(t)
!
+ Pe(t), (3.37)
d
dt
ω̂ropt(t) , k3
ωr(t) − ω̂ropt(t)
, (3.38)
d
dt
k̂opt(t) , k4(k0
opt − k̂opt(t)) + ωr(t)2
ωr(t) − ω̂ropt(t)
, (3.39)
where k̂opt(t) and ω̂ropt(t) are estimations of kopt and ωropt(Vw(t)), respectively. The
feedback gains k1, k2, k3, and k4 are designed as the conditions in Theorem 2 and
J k1 ≥ 0. (3.40)
Lemma 6. In the optimal power control region D, k̂opt(t) is bounded, i.e.,
max k̂opt(t) ≤ k̂opt,ub, (3.41)
k̂opt,ub = 2k−1
4 ω3
rrated + k−1
4

77. ω̂ropt(0)

80. ω2
rrated +

83. k̂opt(0)

86. + k0
opt. (3.42)
Proof. See B.6 for the proof of Lemma 6.
Theorem 2. In addition to Assumption 5, we suppose that ωrref (3.36) for the RSC
control (3.31)-(3.32) is restricted within the optimal control region D as
(ωrref(t), Vw) ∈ D, (3.43)
if there exist positive constants α, v, w and q satisfying



















































Ξ = K
+ K − qI2 0,
2kp − αk̂2
opt,ubξ2
max −
0 1
#
Ξ−1













0
1













− qkd 0,
2ζmin − (wγ + q) ˆ
J − (k3 − k2) − 1 0,
k3 − k2 − ω2
rrated − wγ − q 0,
(2 − vkopt)k4 − ω2
rrated − q 0,
(3.44)
23

87. Fig. 3.3: Control diagram using the adaptive MPPT method.
where
ζmin , min ζ(ωr, Vw), (3.45)
ξ (ωr, ωrref) , ω−1
r ω2
rref + ωr + ωrref, (3.46)
ξmax , max ξ (ωr, ωrref) , (3.47)
ˆ
J , J − k1 0, (3.48)
then, there exists a time to 0 such that for all t ≥ to,

90. ωr(t) − ωropt(Vw(t))

93. 1
√
q
s
1 + ˆ
J−1
w
γ +
k4
ˆ
J−1
v
koptδ2
max. (3.49)
Proof. See B.7 for the proof of Theorem 2.
From Theorem 2, if the reference rotor speed ωrref for the RSC control (3.31)-
(3.32) is designed as (3.36), the rotor speed ωr of the wind turbine will converge to
the neighbor of ωropt and hence, Pm will approach its maximum value. Hence, we
have control diagram as Fig.3.3.
3.4 Comparison of two proposed MPPT schemes
In Subsection 3.2.2 and Subsection 3.3.2, we proposed two MPPT schemes inde-
pendently. The main aim of both two schemes makes the mechanical power Pm of
the wind turbine approach its maximum value. Table 3.1 is the comparison of two
proposed schemes. From this table, when we do not exactly kopt we must use the
24

94. adaptive MPPT method and when we know exactly kopt we can use one of these
methods.
Table 3.1: Comparision of two MPPT methods
Improved MPPT method Adaptive MPPT method
Main objective Pm → Pm max Pm → Pm max
Method Adjust the electric power Pe Adjust the rotor speed ωr
Reference signal Peref ωrref
Requiring RSC controller Pe → Peref ωr → ωrref
Parameter know kopt unknow kopt
25

95. 26

96. Chapter 4
Simulation and Discussions
In this chapter, we evaluate the performance of the proposed MPPT schemes by
comparing the simulation results for the 1.5 MW DFIG wind turbine with the con-
ventional MPPT-curve method with traditional PI control [12]. In all simulations,
we used the parameters of generator [25] and the wind turbine as shown in Table
4.1. We used the power coefficient (3.4) by setting β = 0. Then,
kopt = 1.2467 × 105
kg·m2
, λopt = 6.7562.
The optimal control region D is defined by
ωrmin = 1.15 rad/s, ωrrated = 2.3 rad/s,
Vwmin =
Rωrmin
λopt
=
35.25 × 1.15
6.7562
= 6 m/s,
Vwrated = 12m/s.
The wind speed profile Fig.4.2 we used satisfied Vw(t) Vwrated, and

99. d
dt
Vw(t)

102. ≤ 0.44
m/s2
. Hence,
γ =
λopt0.44
R
=
6.7562 × 0.44
35.25
= 0.0843.
In D, λmax = 10.239 and
λmin =
Rωrmin
Vwrated
=
35.25 × 1.15
12
= 3.4 p.u.
27

103. Table 4.1: Parameters in simulations (DFIG[25] and wind turbine)
Name Symbol Value Unit
Rated power P 1.5 MW
The length of blade R 35.25 m
Normal rotor speed ωrrated 22 rpm
Minimum rotor speed ωrmin 11 rpm
Rated wind speed Vwrated 12 m/s
Rated stator voltage Vs 690 V
Rated rotor voltage Vr 120 V
Rated stator frequency f 50 Hz
Minimum rotor speed Ωr min 1200 rpm
Rated rotor speed Ωrrated 1750 rpm
Number of pole pairs pn 2 p.u
Stator winding resistance Rs 2.65 mΩ
Rotor winding resistance Rr 2.63 mΩ
Stator winding inductance Ls 5.6438 mH
Rotor winding inductance Lr 5.6068 mH
Magnetizing inductance Lm 5.4749 mH
Gearbox ratio N 79.545 p.u
Inertia of system J 445000 kg·m2
28

104. (a)
(b)
(c)
Fig. 4.1: ζp(ωr, Vw) (a),
δ3
δ2
(b), and ζ(ωr, Vw).
29

105. 4.1 Parameters for improved MPPT method
The reference value setting for the RSC from (3.25), with Qsref(t) = 0 are
P =










2 0
0 2










. (4.1)
We used Peref for the RSC control from (3.27) with α = 0.3J. We have
max δ1(t) =
λmax
ηω2
rmin
=
13.4
3.12 × 105 × 1.152
≈ 0 (4.2)
where η = J − α = 0.7J = 3.12 × 105
kg · m2
. With 5m/s ≤ Vw(t) ≤ 12m/s
and 1.15rad/s ≤ ωr(t) ≤ 2.3rad/s, ζp versus ωr and Vw are shown in Figure 4.1a.
From Figure 4.1a, we can obtain min ζp(t) = 0.5 × 105
; we choose χ = 1. With
χ = 1, δ3
δ2
versus ωr and Vw are shown in Figure 4.1b. From that figure, we obtain
∆3 = 0.5273. Obviously, the condition in (3.33) is satisfied, and (3.29) gives the
bound of the tip-speed ratio λ as

108. λ(t) − λopt

111. ≤ ∆3 = 0.5273,
or

114. ωr(t) − ωropt

117. ≤ ∆3
Vrated
R
= 0.1795rad/s.
4.2 Parameters for adaptive MPPT method
The reference values setting for the RSC control (3.31) with irdref = 401.4A,
K = 200I2.
Note that as ird(t) → 401.4 A, the DFIG will generate with a unity power factor.
Here, we used ωrref(t) with
k1 = 0.3J, ˆ
J = J − k1 = 0.7J = 3.12 × 105
,
k2 = 2 ˆ
J, k3 = k2 + 0.001 ˆ
J, k4 = 10, kd = 0.0029J, kp = 100kd,
ω̂ropt(0) = ωrrated, k0
opt = 124610, δmax = 5kopt × 10−4
,
k̂opt(0) = k0
opt, k̂opt,ub = k0
opt + k0
opt + 3k−1
4 ω3
rrated ≈ 2kopt.
30

118. From (3.23) and (3.46), we have
max ξ (ωr, ωrref) =
ω2
rrated
ωrmin
+ ωrmin + ωrrated =
2.32
1.15
+ 1.15 + 2.3 = 8.05.
Obviously, since the power coefficient (3.4) satisfies the condition (3.19) in Lemma
3, ζ(ωr, Vw) is always positive, as shown in Fig. 4.1c. From this figure,
min
(ωr,Vw)∈D
ζ(ωr, Vw) = 1.271 × 105
.
Hence, when v = 0.6 × 10−5
, w = 4.825, α = kdk̂−2
opt,ubξ−2
max and q = 0.4
Ξ = K
+ K − qI2 = diag (399.6, 399.6) 0,
0 1
Ξ−1










0
1










+ qkd = 0.0025 + 0.4kd ≈ 0.4kd,
2kp − αk̂2
opt,ubξ2
max −
0 1
Ξ−1










0
1










− qkd ≈ 198.6kd 0,
and
2ζmin − w ˆ
Jγ − (k3 − k2) − 1 ≈ 2.542 × 105
− 0.105w ˆ
J − 0.001 ˆ
J
= (0.8145 − 0.0843w − 0.001) ˆ
J = 0.4068 ˆ
J q ˆ
J,
k3 − k2 − ω2
rrated − wγ = 0.001 ˆ
J − 2.32
− 0.0843w
≈ 3.12 × 102
− 5.6967 = 306.3033 q,
(2 − vkopt)k4 − ω2
rrated = (2 − v × 1.2467 × 105
)k2 − 2.32
= 6.26 − 5.29 = 0.97 q.
It is easily observed that the five inequalities in Theorem 2 are satisfied. Hence, the
upper bound for the rotor speed ωr(t) in Theorem 2 is
|ωr(t) − ωropt(Vw(t))| ≤
1
√
0.4
s
0.0843
4.825
1 +
1
3.12 × 105
!
+
5 × 25 × 10−8
0.6 × 10−5
1.2467 × 105
3.12 × 105
= 0.254rad/s.
4.3 Simulation results and disscusion
For the above DFIG wind turbine, wind profile, and controllers, the simulation re-
sults are shown in Fig. 4.3. Fig. 4.3a argues that with the conventional method, the
31

119. (a)
(b)
Fig. 4.2: Wind speed profile: (a) wind speed and (b) wind acceleration.
error between ωr(t) and ωropt(t) is still quite large, up to 0.3 rad/s. This is unlikely
with the proposed methods, as ωr(t) always approaches ωropt(t) and guarantees that
the |ωr(t) − ωropt(t)| is always very small, below 0.254 rad/s and 0.1795 rad/s, as
Theorem 2 and Theorem 1, respectively. Compraring to the case of the improved
MPPT method, the adaptive method has a better performance, the maximum of
|ωr − ωropt(t)| is below 0.1 rad/s. Consequently, with the adaptive MPPT method,
the power coefficient Cp is virtually maintained around its maximum value Cpmax=
0.4 p.u. during the simulation interval, as displayed clearly by the blue solid line in
Fig. 4.3b. With the improved MPPT method, Cp fails to be maintained around its
maximum value Cpmax= 0.4 p.u. as the wind condition starts to change rappidly but
it is retained quickly, as the red discontinueous line inFig.4.3b. This performance
is hardly seen in the case with the conventional method, because during the interval
32

120. of rapid decrease in the wind speed, the large error in rotor speed, ωr(t) − ωropt(t),
leads to a reduction of Cp to 0.363 p.u., as shown by the black line in Fig.4.3b.
Concerning the mechanical power output of the wind turbine, Fig.4.3c depicts
the error between Pmax and Pm. The figure shows that when the wind velocity varies
dramatically, the error between Pmax and Pm is approximately to zero with the pro-
posed methods, as shown in the blue line and the red broken line. This is mainly
because the power coefficient Cp remains around Cpmax, as shown in Fig.4.3b. In
other words, with the proposed methods, the main objective, which is to have Pm ap-
proach Pmax, is completely achieved. With the conventional method, however, this
goal is not achievable due to the significant decrease in Cp during sudden variations
in wind conditions. Certainly, comparing to the adaptive method, the improved
method still gives a bigger error between Pm and Pmax during the dramatic change
period of the wind.
With the proposed strategies, the total electrical energy output of the generator
is higher than that with the conventional strategy, as shown in Fig.4.3d. This is
mainly because Pm in the case of the proposed methods has a higher value. This
confirms that the quality of the proposed schemes is always better than that of the
conventional one.
To evaluate the quality of the RSC controller for the adaptive MPPT method and
the improved MPPT method, Fig.4.4 is plotted. Fig.4.4a to Fig.4.4c represent for
the adaptive MPPT method while Fig.4.4e stands for the improved MPPT method.
Fig.4.4a indicates that during the simulation interval, k̂opt is always below its max-
imum value, which is estimated as k̂opt,ub = 2kopt. From Fig.4.4b, the gap between
ωropt(t) and its estimation ω̂ropt(t) is quite small, below 0.097rad/s. Fig.4.4b and
Fig.4.4c demonstrate that errors, ωrref(t) − ωr(t) and irdref(t) − ird(t), are quite small,
approximate to zero, as Lemma 5. Likely, Lemma 4 is guaranteed because the error
xr(t) − xPQ(t) is too small comparing to 1.5MW as Fig.4.4d. In other words, the
RSC controllers which designed for the purposes of the adaptive method and the
improved method have qualified performance.
To affirm the qualified performance of two proposed schemes, hereafter, two
cases of wind speed are used for simulation. Simulation results are shown as Fig.4.5
33

121. and Fig.4.6. Obviously, the wind turbine with the proposed methods has better
performance than with the conventional one, in terms of both the power coefficient
Cp and energy output.
34

122. (a)
(b)
(c)
(d)
Fig. 4.3: Simulation results: (a) ωr(t) − ωropt(Vw(t)), (b) power coefficient Cp(λ(t)),
(c) Pmax(t) − Pm(t), and (d) electrical energy output.
35
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123. (a)
(b)
(c)
(d)
Fig. 4.4: Simulation results: (a) ratio k̂opt/kopt and (b) ωropt(t) − ω̂ropt(t), (c) irdref(t) −
ird(t), (d) xr(t) − xPQ(t).
36
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124. (a)
(b)
(c)
(d)
Fig. 4.5: Simulation results: (a) wind speed, (b) power coefficient, (c)error Pmax(t)−
Pm(t), and (d) energy output.
37
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